Thank you for your post. There are a few more questions concerning this and Sylow theorem that maybe you can help me with.

(i) What is the order of every Sylow 5-subgroup of ?
(ii) How many Sylow 5-subgroups does have?

For (i) I think the answer is 5 because
for (ii) I think using one of the Sylow theorems, that the number of Sylow 5-subgroups is of the form and divides the order of , so I guess from this there is either 1 or 6 Sylow 5-subgroups in , so the answer is 6?? Can anyone verify any of this?

(iii) If is a Sylow p-subgroup of a finite group , and if is a Sylow p-subgroup of a finite group , prove that the direct product is a Sylow p-subgroup of the direct product .

Thank you for your post. There are a few more questions concerning this and Sylow theorem that maybe you can help me with.

(i) What is the order of every Sylow 5-subgroup of ?
(ii) How many Sylow 5-subgroups does have?

For (i) I think the answer is 5 because
for (ii) I think using one of the Sylow theorems, that the number of Sylow 5-subgroups is of the form and divides the order of , so I guess from this there is either 1 or 6 Sylow 5-subgroups in , so the answer is 6?? Can anyone verify any of this?

(iii) If is a Sylow p-subgroup of a finite group , and if is a Sylow p-subgroup of a finite group , prove that the direct product is a Sylow p-subgroup of the direct product .

For (iii) i attempted to prove it like follows: Take

Is this the right method? Thanks for any help anyone can provide.

your answers to (i) and (ii) are correct, although you need to explain in (ii) why the number of Sylow 5-subgroups cannot be 1. for that you need to look at your first question about the number

of 5 cycles. for (iii) look at the orders: since P and Q are Sylow p-subgroups we have where then and

(ii) I think using one of the Sylow theorems, that the number of Sylow 5-subgroups is of the form and divides the order of , so I guess from this there is either 1 or 6 Sylow 5-subgroups in , so the answer is 6?? Can anyone verify any of this?

Hi Jason Bourne.

If you are allowed to assume that is simple, then the answer is immediately obvious. If there were just 1 Sylow 5-subgroup, this would be a proper nontrivial normal subgroup of since is simple, there must therefore be more than 1 Sylow 5-subgroup (indeed more than 1 Sylow subgroup of any order).

Thanks. How do you know that if there is a unique Sylow 5-subgroup then this is Normal in ?

Hi Jason Bourne.

Are you aware of the result that for each prime dividing the order of a finite group all Sylow -subgroups of are conjugate to each other? It follows that if has a unique Sylow -subgroup, then that Sylow -subgroup is normal in

Are you aware of the result that for each prime dividing the order of a finite group all Sylow -subgroups of are conjugate to each other? It follows that if has a unique Sylow -subgroup, then that Sylow -subgroup is normal in